Rough Draft Math Book

I'm starting a new book.

Rough Draft Math by Amanda Jansen.

I saw somewhere that this book is a good choice to follow Building Thinking Classrooms.

I'm one chapter in, 

And I believe I have made a good choice!

Some highlights...

"By expressing ourselves, we learn more..."

"getting into the conversation, helps you understand..."

"participating in math class is an opportunity to continue learning, not an obligation to perform..."

"When we learn, we actively work to make sense of an idea that does not make sense to us yet."

"Our use of bumpy language is actually productive struggle in action."

"Rough draft thinking sounds like communicating what we notice, explaining why something does or does not make sense initially, trying on a new way of seeing a relationship, or making new connections and articulating them".

This first chapter brought to mind several things that happened in my classroom this week.

Rough Draft Math might not be so hard to implement as the next step. 

Scenario #1

While reviewing slope, one of my questions referred to parallel slopes.

A couple questions later, it focused on a slope less steep that y=1/2x+8.

A student suggested using -2 as the new slope.

After graphing the two slopes,

A student just blurted out, "Those lines look like they cross at a 90 degree angle."

This prompted a discussion on what the mathematical vocabulary is for two lines that intersect at 90 degrees.

Which then moved the discussion forward to a place I had never anticipated going...

Teaching what slopes make perpendicular lines.

I wrote on the board that a line with a slope of 1/2 is perpendicular to a line with a slope of -2.

I then made a quick decision to offer some thin-slicing problems to see if students could make sense

of what slopes make perpendicular!

Students quickly talked through the first couple of problems,

If a line has a slope of 1/3, the slope of a line perpendicular to it would be ____

If a line has a slope of 1/4, the slope of a line perpendicular to it would be ____

I then asked if they could put into words what they saw was happening.

Many students yelled out, "Keep Change Flip!"

I could see why they said this, they were seeing fractions flippling and the sign was switching from positive to negative in the examples I had given.

With several wondering question from myself,

"What are these fractions called when the numerator switches places with the denominator?"

"What do you mean by 'change'?"

"What's a better word for when a number goes from positive to negative?"

And pretty soon we got it summarized that perpendicular lines have slopes that are opposite reciprocals.

"participating in math class is an opportunity to continue learning, not an obligation to perform..."

"getting into the conversation, helps you understand..."

I had students in the conversation that rarely jump in.

Some students were leaning forward in anticipation of figuring this out.

It was one of those moments in math where you just let the conversation take over.

Scenario #2

"Rough draft thinking sounds like communicating what we notice, explaining why something does or does not make sense initially, trying on a new way of seeing a relationship, or making new connections and articulating them".

On 2/22/22 students could choose from two activities...

One activity was working with fractions.

They had to make all the problems equal to 2.

It was eye-opening for me to see their number sense and how they thought and worked with fractions.

Of course, dividing fractions, seemed to be the biggest stumbling block.

As I listened to students trying to make sense of these problems,

I wondered how we could make this process easier with pictures.

After deciding to go with the concept of pizzas,

We talked through it together as a class, myself included,

How to visualize these problems to figure out how to get the solution of 2.

Ironically, while typing this, I now have noticed another relationship that would have made this so much easier.

We could easily have just divided it by two to find the divisor we needed!  

An example of where there wasn't enough time to finish the conversation!

"When we learn, we actively work to make sense of an idea that does not make sense to us yet."

When Students Take Notes for Their Future Forgetful Self

So this happened today.

A student created review notes.

On her own.

On her own time.

At home.

To make sure she got the main points from that day's lesson.

Wow.

Now if only all my students would see the benefit of this!  

When Everything Falls Into Place

Today we started with a short thin-slicing review.

I gave the students the equation y=1/2x + 8 and everyone had a white board.

The first task was to write an equation parallel to this equation.

The second task, an equation steeper than this equation.

For those few students that put a fraction for the slope that was less than 1/2,

I simply stated that their equation was not steeper than mine.

They sat there stumped.

I assured them that this mistake was one that would help them down the road.

Which got the attention of other students,

Who now wanted to learn from a mistake!

The third task, the one where I would see how deep their understanding really was,

was to write an equation that was less steep.

I knew that I would see negative numbers pop up as the slope.

And this was one of my goals of our review,

To uncover a misconception and unravel it.

To my surprise, only one student in each class put down a negative number for the slope!

I was expecting so many more.

Also to my surprise, students were showing fractional understanding 

As the fractions they were using were mostly less than 1/2.

Wow!

After summarizing what kind of slopes would be less steep.

We dove into the negative slope.

Many students thought that since a negative is less than 1/2 numerically,

As the slope it would have to be less steep.

Another surprise that came out of the lesson

was being able to push their thinking towards what makes perpendicular lines.

When the slope of -2 was given,

We were able to use comparison strategies to figure out what slope makes a line perpendicular to another line.

Score! 

We just moved our thinking to the high school level!

Thin slicing.

Whole group discussions.

Mistakes.

When all mixed together, make a great learning experience!

Thin Slicing As A Way of Life

Once I started focusing on thin-slicing,

Both the students and I have been enjoying the challenge.

We are in our function unit,

Learning about y=mx+b.

Yesterday, we did a series of problems for writing equations in slope intercept form

From just a point and the slope.

We started off easy.

Slope = 3, (2, 4)

Students were using graphs and tables to figure out the equation.

We reviewed how graphs are the most concrete,

How they help us visualize the line and make finding the equation very easy.

I then introduced the most abstract method,

Using the equation and plugging in the pieces.

Now students have three tools in their toolbox - graphs, tables, and equations!

My next problem was a little more difficult

With the slope = -3 and going through the point (4,1)

While some students still kept with the graph or the table,

Many tried the equation method and found the efficiency of using this method!

And at this point, 

As students found success and understanding,

More often than not,

They would ask for a harder problem!  :)

Vertical White Boards (VWBs) Increase Individualized Instruction

So here's what I've noticed 

That I LOVE about using my vertical white boards.

The individualized instruction has totally INCREASED!!!

As I meander around the room,

I am able to address each group's individual misguided thinking.  :)

Making their instruction personalized and timely.

In the past, I would try to address these things with the whole class.

This was rarely as effective as I was hoping.

Not every student needed this additional instruction.

Off task behavior would start to surface.

Plus, the very student(s) that needed this reteaching

Might disengage as the whole group was made aware of this gap in knowledge.

Now, as clarification, if I see the same misunderstanding board after board after board,

At that point, it might be best to address the whole group in the sake of time.

Using their boards as a springboard for the discussion,

Everyone has an iron in the fire for moving their thinking.

This is usually VERY effective for unsticking the stuck groups.

Let's go back to individualizing instruction

And the example that played out in my room yesterday.

The 8th grade standard we were working on was calculating slope.

As I walked my room,

I was finding hurdles that would stop my kids in their tracks.

First, students didn't know how to graph coordinate points.

This totally derailed the concept of checking their work.

So...a quick mini tour of the coordinate grid system was in order.

Second, I was noticing students struggling with subtracting negative numbers.

As this was something I anticipated already in August,

I was surprised that after our teaching at the beginning of the year

The concept was still not being able to be applied with mastery.

Ok.

Nevermind.

I wasn't too surprised.

Students have struggled with subtracting negative numbers since the beginning of time.

But I was disappointed that all our efforts to battle this in August,

Didn't work.

More work is needed.

And another stumbling block popped up with just a handful of students.

Equivalent fractions.

Students were finding the slopes

But if they got 4/8, they were not recognizing it was the same thing as 1/2.

No wonder I was tired by the end of the day!

Not only was our focus on slope,

But I was making decision after decision how to effectively re-teach these concepts 

To help my students move around the gaps in their learning. 

With the quiz over slope today,

Our measuring stick at the end of the lesson was me asking,

"Where is your confidence factor knowing that you get to show me what you know tomorrow?"

For the students that applied themselves,

Reflected on what they didn't know,

And thought through my re-teaching efforts,

They reported higher confidence factors

Than the students who had not been engaged or reflective in their learning.

Thankfully, with how my Building Thinking Classroom is evolving,

This was not that many students.

Most students walked out more confident then they had walked in!

When the Class Moves the Discussion

As I create my math lesson each day,

I know where we are going with the content.

I also have a pretty good idea of where the struggle will happen.

I allow this struggle,

As the learning that follows is super rich.

But then there are days like today,

Where the magic happens.

Today the students took the lesson 

And went farther than I thought we could go in a day.

I didn't even suggest this route.

I just watched it all unfold on the boards.

The lesson started out with me giving them two points.  (9,6) and (3,5).

We had discovered the formula for how to find slope the day before.

Today we were going to thin-slice the concept, starting with positive whole numbers and positive slopes,

Before moving to negative numbers and negative slopes.

However, with just this one problem, 

I could see quickly as one group collaborated around board #3,

That my lesson was going to take a different perspective.

My lesson was going to cover not only how to find slope,

But where that slope is found in a table of values.

How fractions work in a table of values,

Simplifying fractions,

Adding fractions,

How fractions work in y=mx+b,

And how these points are on the graph.

So many connections.

So many relationships.

So much new learning.

So much review of concepts needed to minimize gaps.  

One problem.

One group of students thinking.

One lesson made better by genuine engagement.

When You Just Have 5 Minutes for Thin-Slicing...Take It!

So today we were using Anchor Charts

As a way to help students take notes for their future forgetful self.  

For the last couple weeks students had been using the concepts of slope and y-intercept,

Without even knowing these terms.

We had been referring to them as speed and starting points.

Today was the day we made the bridge over to the correct mathematical vocabulary!

Using only student input, 

We completed the charts to connect our previous learning to a more abstract way of thinking of functions.

In one of my classes, we had a little extra time on our hands,

Enter...thin-slicing.

As a whole group.

I put up a table of values, starting with x at 0 and asked for the equation.

It didn't take them long,

So the next table I created, did not start at 0.

Didn't stump them.

Then I gave only three points, with gaps in between.

Ummm...a little more stumbling, but they got it! 

Finally, I gave them only two points.

This got them at first, but then their learning started to kick in,

And eventually some were able to write it in slope-intercept form.

It was at this point, that I realized that Monday's lesson might be a little too easy for this class,

Based on what I was seeing.

I told the class this.

That I was going to have to make my next lesson more challenging.

What happened next was NOT what I was expecting.

I was expecting groans.

With exclamations of "NOOOO!!!!"

But instead,

The class cheered!

The. Class. Cheered.

They were excited.

Let that sink in.

They were excited for a challenge.

This Building Thinking Classrooms is the best thing since sliced bread in my books.

I absolutely love how my students are responding.

Because in their responses, I can see that they are thinking,

Connecting the dots,

And ultimately learning the material.  

Happy Weekend ya all!!!  :)

Thin-Slicing Hits the Spot!

While I sit here on this beautiful snow day,

I have time to reflect on yesterday's lesson.

It was based around another form of thin slicing.

Students worked at boards to demonstrate understanding of all the representations:

Story in words, 

Graph,

Table,

and Equation

for new car situations.

Each problem described the situation with just one of the representations.

As students progressed through the problems,

(I started with the story in words, then gave out a problem with just the table, 

before allowing them to tackle a problem from just the equation),

I was able to hone in on specific items for each group, such as

--> drawing precise graphs

--> using the table to check your graph

--> making sure direction of the graph is included in the equation

To begin work on fluency,

we traded out the car situations for savings account situations,

before going strictly to the most abstract,

naked numbers with no story attached.

What I heard when I introduced just the naked number problems,

Was students tying it back to a car situation or savings account.

They were adding the story to it to attach it to their original learning!  

Making math accessible is something I strive for every day.

To help do this, visualization is essential.

And yesterday's thin-slicing lesson had it in spades!  :)